Optimization of Supply Chain Network Perspective Environmental Impact based on Fuzzy Mathematical Programming

Table of contents

1.

supply chain. In general, supply chain network design includes determining the locations, numbers and capacities of network facilities and the aggregate material flow between them. Since the end-of-life (EOL) products have significant impact on environment, a considerable part of literature is dedicated to EOL product management. This has created a need to develop models for reverse supply chain (logistics) network design. Reverse supply chain network design problem addresses the number of collection, recovery, recycling and disposal centers needed, their location and capacities and material flows between them.

In the last several years, many studies have been proposed and much research has been performed on the design and optimization of supply chain networks. In one study, Pirkul and Jayaram an [1] studied a multi-commodity, multi-plant, capacitated facility location problem and proposed an efficient heuristic solution to the problem. In the capacitated plant and warehouse location model, customers typically demand multiple units of different products that are distributed to customer outlets from open warehouses that receive these products from several manufacturing plants. The objective function of the model minimizes the sum of the fixed cost of establishing and operating the plants and the warehouses plus the variable cost of transporting units of products from the plants to the warehouses and distributing the products from the warehouses to the customer, to satisfy the multiple demands of the customers. Recently Ilgin and Gupta et al. [2] present a comprehensive review on environmentally conscious manufacturing and product recovery; below we have surveyed some relevant papers on environmental supply chain network design. Timpe and Kallrath [3] considered a multi-site, multi-product production network and presented a general mixed integer linear programming model that combines aspects related to production, distribution and marketing and involves production sites (plants) and sales points. Cakra vastia et al. [4] developed an analytical model of the supplier selection process in designing a supply chain network. The constraints on the capacity of each potential supplier are considered in the process. The objective of the supply chain is to minimize the level of customer Optimization of Supply Chain Network Perspective Environmental Impact based on Fuzzy Mathematical Programming Subrata Talapatra ? & Md. Shakil ? dissatisfaction, which is evaluated by two performance

2. I. Introduction

well-structured supply chain is an important strategic competency that enables firms to be competitive in today's marketplace. Along this important issue, the concern about environmental impact of business activities results in governmental legislations and environmentally conscious consumers. Environmental or green supply chain management can be defined as integrating environmental aspects into supply chain management covering both forward and reverse supply chains from product design to end-of-life management of used products. The ultimate goal is to consider environment in every decision making process across supply chain, especially the strategic level decisions. Supply chain optimization can help define, recommend, and set flexible supply chain strategies based on organization's operations, resources, and other capabilities. Optimization of supply chain network design, as the most important strategic decision in supply chain management, plays an important role in overall environmental and economic performance of the A criteria: (i) price and (ii) delivery lead time. The overall model operates at two levels of decision making: the operational level and the chain level. The operational level concerns decisions related to optimizing the manufacturing and logistical activities of each potential supplier, to meet the customer's requirements. At the chain level, all of the bids from potential suppliers are evaluated, and the final configuration of the supply chain is determined. The structure of the chain de pends on the product speci fic ations and on the customer's order size. An optimal solution in terms of the models for the two levels can be obtained using a mixed -integer programming technique [4,5] presented a multi-phase mathematical programming approach for effective supply chain design. Syarif et al. [6] considered the logistic chain network problem formulate d by the 0-1 mixed integer linear programming problem. The design of the problem involves the choice of the facilities (plants and distribution center s) to be opened and the distribution network de sign, with the goal of satisfying the demand with minimum cost. For the solution method, the spanning tree-based genetic algorithm using Pr?fer number representation is proposed. Sanayeia et al. [7] proposed an integrated approach of multi-attribute utility theory (MAUT) and linear programming (LP) for rating and choosing the best suppliers and defining the optimum order quantities among selected ones in order to maximize total additive utility. Javadi et al. [8] developed a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-objective no-wait flow shop scheduling problem in a fuzzy environment. The proposed model attempted to simultaneously minimize the weighted mean completion time and the weigh ted mean earliness. A numerical example demonstrated the feasibility of applying the proposed model to no-wait flow shop scheduling problem. The proposed model yielded a compromised solution and the decision maker's overall levels of satisfaction.

To overcome the literature gap, this paper proposes a practical, but tractable, multi-objective fuzzy mathematical programming model for optimization of supply chain networking perspective environmental impact problem that is able to (1) consider both economic and environmental objectives in the design of the supply chain network, (2) integrate the design of reverse and forward supply chain networks to avoid the sub-optimality's results from separated design of forward and reverse supply chains, (3) The model allows decision-makers to design the network configuration with the minimum total cost. (4) Handle the epistemic uncertainty in parameters in real cases results from unavailability or incompleteness and imprecise nature of input data. Also, this paper proposes an efficient solution approach that is able to generate both balanced and unbalanced solutions through making a reasonable tradeoff between environmental and economic objectives.

This paper is organized into eight sections. After the introduction, in which some supply chain models are described, the remainder of the paper is structured as follows. In Section 2, problem statement of the proposed supply chain network is introduced. This model is formulated in section 3 and developed an equivalent auxiliary crisp model in section 5. Implementation and evaluation of this proposed model is described in section 6, and section 7 represents the results and discussion. Conclusions are presented in Section 8. As well as finally appendix and references are attached.

3. II. Problem Statement

The concerned integrated supply chain network in this paper is motivated by a real industrial case. The case is a supply chain network of Coca-Cola drinks in Bangladesh that supplies about 80% of domestic demand. The manufacturer has one production plant with about 600 thousand production capacity per one year. In transportation system of supply chain networking consists of environmental impact like, carbon di oxide (co 2 ) that is responsible for the environmental disasters. To overcome this problem proposed a multi-echelon supply chain network that includes both forward and reverse networks is illustrated in Fig. 1. Through forward network the new products manufactured by plants (production centers) are distributed among customer zones. In the reverse network, the used products are shipped to recycling centers through collection/disassembly centers. All demands of customers must be satisfied and all of the returned products from customers must be collected. Also, a predefined percent of demand from each customer is assumed as returned products from corresponding customer. Unavailability or incompleteness of data in real world network optimization problems is an important challenge that imposes a high degree of uncertainty in such problem. The problem is concerned with the uncertain parameters are presented by fuzzy numbers described by their possibility distribution. The possibility distributions are estimated based on current insufficient data and the decision makers' knowledge. The main objective of this integrated supply chain under uncertain conditionincludes the material flow quantities between different facilities with respect to two conflicting objective functions: (1) minimization of total cost and (2) minimization of total environmental impact. ?? =index of candidate location for production centers, i=1, 2, 3, 4??.i j=index of fixed location of customer zones, j=1, 2, 3, 4??.j k=index of candidate location for collection centers, k=1, 2, 3, 4??.k l=index of existing glass recycling centers, l=1, 2, 3, 4??.l m=index of existing plastic recycling centers, m=1, 2, 3, 4??.m b) Parameters ?? ?? = demand of customer zone, j ?? ?? = rate of return percentage from customer zones, j ð??"ð??" ?? = fixed cost of opening production centers, i ð??"ð??" ?? = fixed cost of opening collection centers, k ?? ???? = transportation cost per product unit from plant, i to customer zones, j ?? ???? = transportation cost of per used product unit from customer zone, j to collection center, k ?? ???? = transportation cost of per glass part of used product unit from collection center, k to glass recycling center, l ? ???? = transportation cost of per plastic part of used product unit from collection center, k to plastic recycling center, m ?? ?? = manufacturing cost per unit of product at production center, i ?? ?? = processing cost for per unit of used product at collection center, k ?? ?? = processing cost for per glass part of used product unit at glass recycling center, l ?? ?? = processing cost for per plastic part of used product unit at plastic recycling center, m ?? ?? = maximum capacity of production center, i ?? ?? = maximum capacity of collection center, k ?? ?? = maximum capacity of glass recycling center, l ?? ?? = maximum capacity of plastic recycling center, m ???? ?????? = Environmental impact per production of one unit of product ?? ???? ?????? = environmental impact of shipping one unit of product from plant, i to customer zone, j ?? ???? ?????? = environmental impact of shipping one unit of used product from customer zone, j to collection center, k ?? ???? ?????? = environmental impact of shipping glass part of used product unit from collection center, k to glass recycling center, l ?? ???? ?????? = environmental impact of shipping plastic part of used product unit from collection center, k to plastic recycling center, m ???? ?????? = environmental impact per handling one unit of collected used product at collection centers ???? ?????? = environmental impact of recycling the glass part of one unit of used product ???? ?????? = environmental impact of recycling the plastic part of one unit of used product c) Variables ?? ???? = quantity of product shipped from plant, i to customer zone, j ?? ???? = quantity of used product shipped customer zone, j to collection center, k

?? ???? =+ ? ? (?? ?? + ?? ???? )?? ???? ?? ?? + ? ? (?? ?? + ?? ???? )?? ???? ?? ?? + ? ? (?? ?? + ?? ???? )?? ???? ?? ?? + ? ? (?? ?? + ? ?? ?? )?? ???? ?? ??

Here transportation costs between facilities are calculated by multiplying the transportation cost of one unit shipping per unit of distance.

For the second objective: minimizing the total environmental impact

The purpose of this supply chain network is to fulfill the customer demand by producing and distributing the product at forward network and the safe management of product by reverse network.

The purpose of using ECO-indicator is to estimate the environmental impact of different supply chain network configurations. Following ECO-indicators are considered for this supply chain network design.

? The production (pro)

? Transportation from production centers to customer zone (tpc)

? Transportation from customer zone to collection centers (tcc)

? Handling the used product at collection centers(col) ? Transportation from collection to glass recycling centers (tcs) ? Glass recycling center (src)

? Transportation from collection centers to plastic recycling centers (tcp)

? Plastic recycling centers (src) Min w 2 = ? ? (???? ?????? + ???? ???? ?????? )?? ???? ?? ?? + ? ? (???? ?????? + ???? ???? ?????? )?? ???? ?? ?? + ? ? (???? ?????? + ???? ???? ?????? )?? ???? ?? ?? + ? ? (???? ?????? + ???? ???? ?????? )?? ???? ?? ??

4. IV. Constraints

5. Demand and return satisfaction constraints

Here following constraints (3) and ( 4) ensure the demands of all customers are satisfied and the entire used products are collected from the customer zones.

? ?? ???? ?? ? ?? ??(3)? ?? ???? ?? ? ?? ?? ?? ??(4)

a) Flow Balance Constraints

Here constraints ( 5) and ( 6) ensure the flow balance at collection centers. Two EOL options are considered in the proposed model, the collected used product should be sent to glass and plastic recycling centers. Therefore the total number of plastic and glass parts should be equal to recycling centers because they are disassembled from one used product.

? ?? ???? ?? ? ? ?? ???? ?? (5) ? ?? ???? ?? ? ? ?? ???? ?? (6) ? ?? ???? ?? ? ?? ?? ?? ?? (8) ? ?? ???? ?? ? ?? ??(9)

? ?? ???? ?? ? ?? ?? (10) Here constraints (8) to (10) are capacity constraints on production, collection and glass recycling and plastic recycling centers respectively. Also constraints (7) and ( 8) prohibit the units of new and used products from being transferred to production and collection centers which are not opened respectively.

6. Decision variables constraints

The following constraints are related to the binary and non-negatively restrictions on the corresponding decision variables.

?? ?? , ?? ?? Ñ?"{0,1}(11)

V. Proposed Method This is a multi-objective probabilistic mixed integer programming model. To solve this model a two phase approach is proposed one is the method of Jimenez to convert the proposed model and the second Second objective function: minimization of total environmental impact First objective Function: minimization of total cost VI. Equivalent Auxiliary Crisp Model Jimenez et al. [9] method is selected to develop this equivalent auxiliary crisp model as well as this triangular, trapezoidal and nonlinear ones in both symmetric and asymmetric functions. This method also computational efficient to solve fuzzy linear problems as it can preserve its linearity and do not increase the number of objective functions and inequality constraints. The detail of this method is given in Appendix.

Equivalent auxiliary crisp model can be formulated as follows:

minw 1 = ? ( ð??"ð??" ?? ?????? +2ð??"ð??" ?? ?????? +ð??"ð??" ?? ?????? 4 )?? ?? ?? + ? ( ð??"ð??" ?? ?????? +2ð??"ð??" ?? ?????? +ð??"ð??" ?? ?????? 4 )?? ?? ?? + ? ? ( ?? ?? ?????? +2?? ?? ?????? +?? ?? ?????? +?? ???? ?????? +2?? ???? ?????? +?? ???? ?????? 4 ) ?? ?? ?? ???? + ? ? ( ?? ?? ?????? +2?? ?? ?????? +?? ?? ?????? +?? ???? ?????? +2?? ???? ?????? +?? ???? ?????? 4 ) ?? ?? ?? ???? + ? ? ( ?? ?? ?????? +2?? ?? ?????? +?? ?? ?????? +?? ???? ?????? +2?? ???? ?????? +?? ???? ?????? 4 ) ?? ?? ?? ???? + ? ? ( ?? ?? ?????? +2?? ?? ?????? +?? ?? ?????? +? ???? ?????? +2? ???? ?????? +? ???? ?????? 4 ) ?? ?? ?? ???? minw 2 = ? ? ????? ?????? + ???? ???? ?????? ??? ???? ?? ?? + ? ? ????? ?????? + ???? ???? ?????? ??? ???? ?? ?? + ? ? (???? ?????? + ???? ???? ?????? )?? ???? ?? ?? + ? ? ????? ?????? + ???? ???? ?????? ??? ???? ?? ?? Subject to, ? ?? ???? ?? ? ? ( ?? ?? ?????? +?? ?? ?????? 2 ) + (1-?)( ?? ?? ?????? +?? ?? ?????? 2 ) ? ?? ???? ?? ? ??? ? ?? ?? ?????? +?? ?? ?????? 2 ? + (1 ? ??)( ?? ?? ?????? +?? ?? ?????? 2 )? ??? ? ?? ?? ?????? +?? ?? ?????? 2 ? + (1 ? ??)( ?? ?? ?????? +?? ?? ?????? 2 )? ? ?? ???? ?? = ? ?? ???? ?? ? ?? ???? ?? = ? ?? ???? ?? ? ?? ???? ?? ??? ?? ??? ? ?? ?? ?????? +?? ?? ?????? 2 ? + (1 ? ??)( ?? ?? ?????? +?? ?? ?????? 2 )? ? ?? ???? ?? ??? ?? ??? ? ?? ?? ?????? +?? ?? ?????? 2 ? + (1 ? ??)( ?? ?? ?????? +?? ?? ?????? 2 )? ? ?? ???? ?? ???? ? ?? ?? ?????? +?? ?? ?????? 2 ? + (1 ? ??)( ?? ?? ?????? +?? ?? ?????? 2 )? ? ?? ???? ?? ???? ? ?? ?? ?????? +?? ?? ?????? 2 ? + (1 ? ??)( ?? ?? ?????? +?? ?? ?????? 2 )? ?? ?? , ?? ?? Ñ?"{0,1}

?? ???? , ?? ???? , ?? ???? , ?? ???? ? 0

7. VII. Implementation and Evaluation

The validity of the developed model as well as the usefulness of the proposed solution method is investigated via the data withdrawn from the case study. The manufacturer firm has nine customer zones. The firm is responsible to collect the used product from domestic customers therefore the return rate from the foreign customer is considered equal to zero. To estimate the possibility of distribution parameters first objective data is gathered and the firm managers determined three prominent values (most likely, most pessimistic and most optimistic) of triangular fuzzy numbers according to available data. The fuzzy data for demand and rate of return each customer is represented in table: 1 for the over three years.

8. VIII. Results and Discussion

Firm supplies products from different production centers to customer's zone as well as shipped using transportation by trucks. Products manufactured in production centers are directly dispatched to customer zone, and the manufacturer has to pay transportation costs. The firm assigns trucks with respect to the capacities of truck options and transports the products from the production center to the customer zone.

Table 4. presents the transportation cost form production center to customer zone; here trucks are used to transport the products. Table 10. Represents the transportation cost of product from collection center to plastic recycling center by using trucks in reverse supply chain networking. Table 12. Represents the environmental impact of shipping product from production center to customer zone, here environmental impact means the amount of carbon di oxide (CO 2 ) obtained from the trucks during transportation. Environmental impact per production of one unit of product, ???? ?????? =42 ? ?(???? ?????? + ?? ???? ?????? ) = 6155 ?? ?? Table 13. Represents the environmental impact of shipping one unit of product from customer zone to collection center, here environmental impact means the amount of carbon di oxide (CO 2 ) obtained from the trucks during transportation. Environmental impact of handling one unit of collected used product at collection center, ???? ?????? =32 ? ?(???? ?????? + ?? ???? ?????? ) = 5416 ?? ?? Table 14. Represents the environmental impact of shipping glass part from collection center to glass recycling center, here environmental impact means the amount of carbon di oxide (CO 2 ) obtained from the trucks during transportation. Environmental impact of recycling one unit of glass part, ???? ?????? =40 ? ?(???? ?????? + ?? ???? ?????? ) = 3197 ?? ?? Table 15. Represents the environmental impact of shipping plastic part from collection center to plastic recycling center, here environmental impact means the amount of carbon di oxide (CO 2 ) obtained from the trucks during transportation. The above solution represent the minimization of total cost is 0.1394333E+11; here no iteration is required to get the optimal solution. The optimal solution is obtained for the proposed supply chain networking contains of variables of production centers (X) is 10175.00 that shows that if a new production center is opened than cost will increase otherwise reduced amount is 10175.00. The variables (Y) represent the collection center that is obtained 15925.00, that presents if a collection center is opened than cost will increase amount of 15925.00 otherwise reduced.

Variables (Z) show the quantity of product shipped from production centers (i) to customer zone (j) that is obtained 15012.00 units for the minimization of cost.

Variables (W) shows the quantity of product shipped from customer zone (j) to collection center (k) that is obtained 10730.61 for the minimization of total cost. For the reverse flow variables (m) & (n) presents the quantity of used product shipped from collection center (k) to glass recycling center (l) & quantity of plastic part of used product shipped from collection center (k) to plastic recycling center (m) those are 43925.00 and 41375.00 reduced cost. Inequality constraint to transform it to equality slack and surplus values for the row 1,2,4,5 are 0.1394333E+11, 7281.000, 6736.500 and row 2 & 3 presents the transportation cost of production center(i) to customer zone(j)& customer zone (j) to collection center (k). The above solution represent the minimization of total environmental impact here environmental impact minimization means the reduction of carbon di oxide (CO 2 ) during the transportation of product from production center (i) to customer zone (j) and customer zone (j) to location centers (k) finally location centers (l) to glass or plastic recycle center(l or m) through trucks.

Here for the proposed supply chain networking problem only carbon di oxide (CO 2 ) is considered as an environmental impact others are neglected. The 2 nd objective function shows the minimization of environmental impact that is 0.9239886E+08 as well as no iteration is required to get the optimal solution. Variables (Z) show the quantity of product shipped from production centers (i) to customer zone (j) that is obtained 15012.00 units for the minimization of environmental impact. The variables (Y) represent the collection center that is obtained reduced 5416.000. A variable (M) is the quantity of glass part shipped from collection center (k) to glass recycling center (l) than the reduced amount of 3197.000. A variable (N) is the quantity of plastic part shipped from collection center (k) to plastic recycling center (m) than the reduced amount of 1510.000. A variable (W) is the quantity of used product shipped from customer zone (j) to collection center (k) amount of 10730.61.Inequality constraint to transform it to equality slack and surplus values for the row 1,4,5 are 0.9239886E+08, 7281.000, 6736.500 and dual prices are showing in row 1 & 2.

9. IX. Conclusion

Effective supply chain network design and optimization of the network are tasks that provide a competitive advantage to firms and organizations in today's highly intractable global business environment. In this study, design and optimization supply chain networking based on multi-objective fuzzy mathematical programming model, this consists of minimizing the total cost and environmental impact and determining the optimal physical shipment of product from production center to customer zone in forward flow and collection center to recycling center in reverse flow. The proposed fuzzy model includes the design of the network configuration with a minimum total cost and environmental impact under the fuzzy capacity constraints with triangular and trapezoidal member ship functions. The total cost involves the following: the transportation costs between production center and customer zone; customer zone to collection center and collection center to recycling center. To solve the proposed optimization model, an interactive fuzzy solution approach is developed based on the econstraint method and the possibility programming approach proposed by Jimenezet al. [9].The proposed hybrid solution approach is able to generate both balanced and unbalanced solutions and making a reasonable tradeoff between environmental and economic objectives. The effectiveness of the developed fuzzy optimization model as well as the usefulness of the proposed solution approach is investigated through a real industrial case. Finally, a sensitivity analysis developed to show the correlation between the objective function value and the constraints using LINDO 12 optimization software.

According to the ranking method of Jimenez [10], for any pair of fuzzy numbers 'a and b', the degree in which a is bigger than b can be de fined as follows.

When ? m (a,b)?? it will be said that a is bigger than, or equal to, b at least in degree of ? and it will be represented as a? ? b. Now, consider the following fuzzy mathematical programming model in which all parameters are defined as triangular or trapezoidal fuzzy numbers.

Min z= ? t x ST, ?? ?? ?? ? ?? ?? , i= 0,1,2???l;

Eq. ( 17) can be rewritten as follows.

?(1 ? ??)??

Also, Jimenez et al. [9] showed that a feasible solution like x 0 is an acceptable optimal solution of the model (18)

Figure 1. Fig. 1 :
1Fig. 1 : Concerned integrated supply chain network III. Model Formulation The indices, parameter and variables used to formulate the concerned environmental supply chain network design problem. a) Indices
Figure 2. 2 =
2?? ?????? +2?? ?????? +?? ?????? 4
Figure 3.
1, if a production center opened at location, ?? ?? ?? =? 0, otherwise
1, if a collection center opened at location, ?? ?? ?? =? 0, otherwise
d) Objective Function
There are two objective functions are
considered:
?? ???? = quantity of plastic part of used product shipped from collection center, k to plastic recycling center, m i. Minimization of total cost ii. Minimization of total environmental impact
Minw 1 =? ð??"ð??" ?? ?? ?? ?? + ? ð??"ð??" ??
Note: ?? ?? ?? + ? ? (?? ?? + ?? ???? )?? ???? ?? ??
Figure 4.
Optimization of Supply Chain Network Perspective Environmental Impact based on Fuzzy
Mathematical Programming
b) Capacity Constraint
? ?? ????
Note: ??? ?? ?? ?? ??
Figure 5. Table 1 :
1
different kind of membership functions such as
model is based on mathematical concepts that is
expected interval and expected value of fuzzy numbers
and also explain a ranking method which can support
Figure 6. Table 2 :
2
Location(i) Fixed Cost, ?? ?? (Thousand) pes mos opt Capacity, ?? ?? (Thousand) pes mos opt
Khulna 13300 14500 15300 190 200 210
Rajshahi 13500 14700 15400 190 200 210
Narayongonj 13600 14800 15500 200 210 220
Chitagong 13500 14700 15400 165 180 195
Dhaka 13000 14000 15000 190 200 210
Rangpur 13600 14700 15400 190 200 210
Barisal 13400 14200 15200 165 180 195
Joshor 0 0 0 170 190 210
Figure 7. Table 3 :
3
Location, i Fixed cost, ?? ?? (Thousand) pes mos opt Capacity, ?? ?? (Thousand) pes mos opt
Khulna 1700 1740 1780 240 245 250
Rajshahi 1750 1790 1830 240 245 250
Chitagong 1700 1740 1780 250 255 260
Dhaka 1680 1720 1740 220 225 230
Narayongonj 1780 1830 1880 230 235 240
Rangpur 1760 1810 1860 220 205 210
Savar 1740 1780 1820 200 205 210
Barisal 1720 1750 1780 210 215 220
Joshor 1730 1770 1810 225 230 235
Figure 8. Table 4 :
4
Production Customer Center, i 1 2 3 4 5 6 7 8
Zone, j
pes 900 1000 900 1000 1100 1100 1000 1000
1 mos 1000 1200 1200 1200 1300 1250 1400 1250
opt 800 1100 800 1100 1200 1000 1200 1200
pes 1100 1100 1000 1100 1000 1200 1000 1200
2 mos 1200 1350 1100 1250 1400 1250 1300 1400
opt 1150 1200 1200 1300 1200 1300 1200 1100
pes 1200 1200 1100 1100 1200 1400 1000 1500
3 mos 1400 1400 1150 1300 1450 1100 1350 1600
opt 1100 1100 1200 1200 1000 1200 1200 1400
pes 1400 1200 1000 1100 1200 1200 1300 1300
4 mos 1500 1500 1200 1350 1500 1150 1400 1500
opt 1300 1300 900 1200 1400 1000 1200 1200
pes 1200 1200 1000 1200 1300 1000 1000 1100
5 mos 1350 1550 1300 1400 1500 1200 1500 1700
opt 1100 1100 1200 1100 1400 1100 1300 1200
pes 1200 1400 1300 1200 1400 1200 1200 1300
6 mos 1500 1600 1350 1450 1600 1250 1550 1750
opt 1100 1350 1200 1100 1300 1300 1200 1200
pes 1400 1200 1500 1200 1400 1200 1100 1400
7 mos 1600 1700 1400 1500 1250 1300 1600 1600
opt 1300 1300 1300 1100 1500 1000 1200 1300
pes 1500 1400 1400 1300 1000 1000 1000 1200
8 mos 1700 1750 1450 1600 1300 1350 1650 1550
opt 1600 1500 1200 1400 1200 1200 1100 1300
pes 1500 1300 1000 1000 1200 1000 1200 1000
9 mos 1650 1400 1500 1250 1350 1400 1400 1600
opt 1400 1200 900 1200 1100 1200 1000 1200
Figure 9. Table 5 .
5
Figure 10. Table 5 :
5
Production pes mos opt
centers, i
1 10000 10500 11000
2 10500 12000 12500
3 11000 11500 10000
4 10000 12000 11000
5 11500 12500 13000
6 12000 11000 14000
7 11000 10000 15000
8 10500 11500 12000
Figure 11. Table 6 .
6
Figure 12. Table 6 :
6
Collection
Center, k
1 2 3 4 5 6 7 8 9
Customer
Zone, j
pes 650 800 900 650 700 650 600 700 800
1 mos 800 900 1000 700 600 700 800 900 1000
opt 700 700 800 600 800 600 700 600 900
pes 500 700 700 600 600 700 700 800 650
2 mos 750 800 600 650 750 800 850 900 700
opt 600 750 800 700 700 750 800 700 600
pes 700 700 600 700 600 700 600 750 750
3 mos 800 600 700 900 1000 800 700 800 900
opt 850 750 500 750 900 900 650 700 800
pes 900 1000 400 800 800 800 800 700 700
4 mos 1000 1200 800 900 700 600 1000 800 900
opt 800 900 500 850 850 900 900 600 800
pes 700 800 600 900 900 800 700 850 1000
5 mos 800 900 700 650 600 700 800 900 950
opt 600 750 750 950 800 900 900 950 800
pes 800 800 850 700 900 700 800 700 550
6 mos 850 950 750 900 700 600 550 500 600
opt 700 900 800 800 800 950 900 600 700
pes 750 800 900 700 900 700 800 700 550
7 mos 800 900 950 900 700 600 550 500 600
opt 700 700 800 800 800 950 900 600 700
pes 800 900 700 1000 750 1000 500 600 600
8 mos 900 950 800 850 800 700 600 650 550
opt 700 800 750 900 700 900 700 700 700
pes 600 700 1100 700 750 700 500 850 650
9 mos 900 800 1000 800 800 800 600 900 700
opt 500 600 900 600 650 600 550 800 600
Figure 13. Table 7 .
7
Figure 14. Table 7 :
7
Collection pes mos opt
center, k
1 1000 1200 1100
2 800 700 1000
3 900 800 1200
4 1100 1000 900
5 800 700 1000
6 1100 800 1000
7 1000 900 1200
8 1100 800 1000
9 1000 900 1200
Figure 15. Table 8 .
8
Figure 16. Table 8 :
8
Glass recycling
center, l 1 2 3 4
collection
center, k
pes 500 300 900 400
1 mos 600 400 500 300
opt 400 200 600 500
pes 600 400 600 400
2 mos 700 750 650 600
opt 650 500 700 300
pes 450 600 700 500
3 mos 500 550 600 700
opt 550 700 750 600
pes 700 800 600 500
4 mos 650 700 800 700
opt 600 900 700 600
pes 500 800 700 700
5 mos 600 500 450 300
opt 400 700 600 800
pes 400 400 500 700
6 mos 450 550 650 750
opt 300 500 550 600
pes 500 600 900 500
7 mos 800 750 700 850
opt 600 700 800 300
pes 700 650 500 200
8 mos 900 600 300 400
opt 800 700 600 300
pes 400 700 500 400
9 mos 300 450 400 500
opt 500 800 600 600
Figure 17. Table 9 .
9
Figure 18. Table 9 :
9
Glass
recycling pes mos opt
center, l
1 500 600 900
2 800 900 700
3 700 450 400
4 600 650 500
Figure 19. Table 10 :
10
Plastic recycling
center, m
Collection 1 2 3 4
Center, k
pes 300 400 400 400
1 mos 400 300 450 500
opt 500 500 500 300
Figure 20. Table 11 .
11
Figure 21. Table 11 :
11
m pes mos opt
1 500 600 700
2 600 650 550
3 500 400 450
4 500 650 700
Figure 22. Table 12 :
12
Year 2014
IV Version I
( ) G Volume XIV Issue
?????? ) of shipping product from production center,
i to customer zone, j
i 1 2 3 4 5 6 7 8
j
1 30 32 34 30 32 36 38 40
2 42 35 36 38 39 40 42 45
3 46 47 48 39 40 46 47 48
4 50 52 42 46 47 48 49 50
5 52 53 54 55 46 48 50 52
6 54 32 34 36 38 40 42 45
7 46 48 50 52 54 46 48 50
8 42 46 40 38 35 45 46 48
9 44 48 38 36 32 40 48 46
Figure 23. Table 13 :
13
j to collection center, k
J 1 2 3 4 5 6 7 8 9
k
1 24 26 28 30 39 32 36 38 40
2 26 28 30 32 34 36 38 40 38
3 32 36 38 44 32 24 28 30 32
4 34 34 40 42 34 32 30 28 34
5 30 28 32 36 34 35 28 30 32
6 36 32 38 40 36 26 32 48 38
7 38 34 36 38 38 32 36 46 36
8 36 36 34 36 36 44 38 44 32
9 40 38 32 36 40 42 40 42 34
Figure 24. Table 14 :
14
l 1 2 3 4
k
1 50 48 50 52
2 52 46 54 54
3 48 44 56 55
4 46 42 58 42
5 44 44 42 44
6 50 46 44 46
7 52 48 46 48
8 54 50 48 50
9 50 52 50 52
Figure 25. Table 15 :
15
m 1 2 3 4
k
1 16 18 20 22
2 24 18 22 24
3 26 20 18 16
4 28 18 22 18
5 30 28 18 18
6 32 26 20 20
7 34 24 16 22
8 32 22 20 24
9 18 20 18 18
Environmental impact of recycling one unit of plastic product, ???? ?????? =20
? ?(???? ?????? + ?? ???? ?????? ) = 1510
?? ??
Figure 26. Table 16 .
16
Figure 27. Table 16 :
16
l, m pes ?? ?? mos opt pes ?? ?? mos opt
1 100 150 200 180 150 200
2 200 150 180 180 200 250
3 190 200 250 220 180 250
4 230 250 180 240 250 180
Simplifications of the constraints are obtained by developing a program using Code blocks programming
software:
minw 1 = Constraints,
?? ???? ? 15012
?? ???? ? 10730.61
?? ???? ? 13860 or ?? ???? ? 0
?? ???? ? 18639 or ?? ???? ? 0
?? ???? ? 7281
?? ???? ? 6736.5
Figure 28. Table 17 :
17
Symbol Modified
Symbol
?? ?? X
?? ?? Y
?? ???? Z
?? ???? W
?? ???? M
?? ???? N
Figure 29.
1
2
3

Appendix A

Appendix A.1 Appendix

The Jimenez et al. [9] method is based on the definition of the ''expected interval'' and the ''expected value'' of a fuzzy number. Assume that ? is a triangular fuzzy number. The following equation can be defined as the membership function of ?.

Here ?? ?????? , ?? ?????? ?????? ?? ?????? are the three prominent points (the most likely, the most pessimistic and the most optimistic values), respectively. Eqs. ( 13) And ( 14) define the expected interval (EI) and the expected value (EV) of triangular fuzzy number ?.

Appendix B

  1. A two-stage model for the design of supply chain networks. A Cakravastia , I S Toha , N Nakamura . International Journal of Production Economics 2002. 80 (3) p. .
  2. A two-level network for recycling sand: A case study. A I Barros , R Dekker , V Scholten . Eur. J. Oper. Res 1998. 110 p. .
  3. An integrated group decision-making process for supplier selection and order allocation using multi-attribute utility theory and linear programming. A Sanayeia , S F Mousavib , M R Abdic , A Mohagharb . Journal of the Franklin Institute 2008. 345 p. .
  4. Study on multi-stage logistic chain network: a spanning tree-based genetic algorithm approach. A Syarif , Y S Yun , M A Gen . Computers & Industrial Engineering 2002. 43 p. .
  5. No-wait flow shop scheduling using fuzzy multi-objective linear programming. B Javadi , M Saidi-Mehrabad , A Haji , I Mahdavi , F Jolai , N Mahdavi-Amiri . Journal of the Franklin Institute 2008. 345 p. .
  6. Optimal planning in large multi-site production networks. C H Timpe , J Kallrath . European Journal of Operational Research 2000. 126 (2) p. .
  7. A bi-objective reverse logistics network analysis for post-sale service. F Du , G W Evans . Comput. Oper. Res 2008. 35 p. .
  8. A genetic-based heuristic for the dynamic integrated forward/reverse logistics network for 3PLs. H J Ko , G W Evans . Comput. Oper. Res 2007. 34 p. .
  9. A multi-commodity, multiplant, capacitated facility location problem: formulation and efficient heuristic solution. H Pirkul , V Jayaraman . Computers & Operations Research 1998. 25 (10) p. .
  10. Reverse logistic network re-design for copiers. H R Krikke , A Van Harten , P C Schuur . OR Spektrum 1999. 21 p. .
  11. Network model and optimization of reverse logistics by hybrid genetic algorithm. J E Lee , M Gen , K G Rhee . Comput. Ind. Eng 2009. 56 p. .
  12. The theory and practice of reverse logistics. L Meade , J Sarkis , A Presley . Int. J. Logistics Syst. Manage 2007. 3 p. .
  13. Environmentally conscious manufacturing and product recovery (ECMPRO): a review of the state of the art. M A Ilgin , M Surendra , S M Gupta . J. Environ. Manage 2010. 91 p. .
  14. A stochastic model for forward_reverse logistics network design under risk. M El-Sayed , N Afia , A El-Kharbotly . Comput. Ind. Eng 2010. 58 p. .
  15. Ranking fuzzy numbers through the comparison of its expected intervals. M Jimenez . Int. J
  16. Linear programming with fuzzy parameters: An interactive method resolution. M Jimenez , M Arenas , A Bilbao , M V Rodriguez . Eur. J. Oper. Res 2007. 177 p. .
  17. A possibilistic programming approach for closed -loop supply, M S Pishvaee , S A Torabi .
  18. A stochastic optimization model for integrated forward/reverse logistics network design. M S Pishvaee , F Jolai , J Razmi . J. Manufact. Syst 2009. 28 p. .
  19. A memetic algorithm for bi-objective integrated forward/reverse logistics network design. M S Pishvaee , R Z Farahani , W Dullaert . Comput. Oper. Res 2010. 37 p. .
  20. Saldanha-da-Gama, Facility location and supply chain management -A review. M T Melo , S Nickel , F . Eur. J. Oper. Res 2009. 196 p. .
  21. Locating collection centers for distance-and incentive-dependent returns. N Aras , D Aksen . Int. J. Prod. Econ 2008. 111 p. .
  22. Optimization of Supply Chain Network Perspective Environmental Impact based on Fuzzy Mathematical Programming,
  23. Green supply-chain management: a state-of-the-art literature review. S K Srivastara . Int. J. Manage. Rev 2007. 9 (1) p. .
  24. A multi-phase mathematical programming approach for effective supply chain design. S Talluri , R C Baker . European Journal of Operational Research 2002. 141 (3) p. .
  25. A closed-loop logistics model for manufacturing. V Jayaraman , V D R GuigeJr , R Srivastava . J. Oper. Res. Soc 1999. 50 p. .
Notes
1
© 2014 Global Journals Inc. (US)
2
Optimization of Supply Chain Network Perspective Environmental Impact based on FuzzyMathematical Programming
3
© 2014 Global Journals Inc. (US)Global Journal of Researches in Engineering
Date: 2014-01-15